Triangle $ABC$ has $AC = 5$. $D$ and $E$ are on side $BC$ such that $AD$ and $AE$ trisect $\angle BAC$, with $D$ closer to $B$ and $DE =\frac32$, $EC =\frac52$ . From $B$ and $E$, altitudes $BF$ and $EG$ are drawn onto side $AC$. Compute $\frac{CF}{CG}-\frac{AF}{AG}$ .

Let $AD$ be a diameter of a circle of radius $r,$ and let $B,C$ be points on the circle such that $AB=BC=\frac r2$ and $A\neq C.$ Find the ratio $\frac{CD}{r}.$

Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent. In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles. Find the radius of $C_4$. [asy] unitsize(0.4 cm); pair[] O; real[] r; O[1] = (-12/5,16/5); r[1] = 1; O[2] = (0,5); r[2] = 2; O[3] = (0,0); r[3] = 3; O[4] = (-132/115, 351/115); r[4] = 6/23; draw(Circle(O[1],r[1])); draw(Circle(O[2],r[2])); draw(Circle(O[3],r[3])); draw(Circle(O[4],r[4])); label("$C_1$", O[1]); label("$C_2$", O[2]); label("$C_3$", O[3]); [/asy]

Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.

The diagonal of square I is $a+b$. The perimeter of square II with [i]twice[/i] the area of I is: $ \textbf{(A)}\ (a+b)^2\qquad\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad\textbf{(C)}\ 2(a+b)\qquad\textbf{(D)}\ \sqrt{8}(a+b) \qquad$ $\textbf{(E)}\ 4(a+b) $

Let $ABCD$ be a convex quadrilateral such that your diagonals $AC$ and $BD$ are perpendiculars. Let $P$ be the intersection of $AC$ and $BD$, let $M$ a midpoint of $AB$. Prove that the quadrilateral $ABCD$ is cyclic, if and only if, the lines $PM$ and $DC$ are perpendiculars.

What is the largest number of regions into which a plane can be divided by drawing $n$ pairs of parallel lines?

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

Let $ABC$ be an acute triangle and $G$ be the intersection of the meadians of triangle $ABC$. Let $D $be the foot of the altitude drawn from $A$ to $BC$. Draw a parallel line such that it is parallel to $BC$ and one of the points of it is $A$.Donate the point $S$ as the intersection of the parallel line and circumcircle $ABC$. Prove that $S,G,D$ are co-linear [asy] size(6cm); defaultpen(fontsize(10pt)); pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C); draw(A--B--C--cycle, black+linewidth(1)); draw(A--S^^A--D, magenta); draw(S--D, red+dashed); draw(circumcircle(A, B, C), heavymagenta); string[] names = {"$A$", "$B$", "$C$","$D$", "$G$","$S$"}; pair[] points = {A, B, C,D,G,S}; pair[] ll = {A, B, C,D, G,S}; int pt = names.length; for (int i=0; i<pt; ++i) dot(names[i], points[i], dir(ll[i])); [/asy]

Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.

Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the midpoints of sides $\overline{AB}$ and $\overline{AC}$ respectively. Let $F$ be the point such that $D$ is the midpoint of $\overline{EF}$. Let $\Gamma$ be the circumcircle of triangle $FDB$. Let $G$ be a point on the segment $\overline{CD}$ such that the midpoint of $\overline{BG}$ lies on $\Gamma$. Let $H$ be the second intersection of $\Gamma$ and $FC$. Show that the quadrilateral $BHGC$ is cyclic. \\ \\ [i]Proposed by Art Waeterschoot.[/i]

The line $\ell$ passes through vertex$ B$ and the interior of regular hexagon $ABCDEF$. If the distances from $\ell$ to the vertices $A$ and $C$ are $7$ and $4$, respectively, compute the area of hexagon $ABCDEF$.

A beam of length $ a $ is suspended horizontally with its ends on two parallel ropes equal $ b $. We turn the beam through an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam rise?

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.

[u]Round 5[/u] [b]p13.[/b] You have a $26 \times 26$ grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of $2 \times 2$ squares that are entirely red? [b]p14.[/b] Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats? [b]p15.[/b] Let $n \ge 1$ be an integer and $a > 0$ a real number. In terms of n, find the number of solutions $(x_1, ..., x_n)$ of the equation $\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2$ such that $x_i$ belongs to the interval $[0, a]$ , for $i = 1, 2, . . . , n$. [u]Round 6 [/u] [b]p16.[/b] All roots of $$\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0$$ are written in the form $r(\cos \phi + i\sin \phi)$ for $i^2 = -1$, $r > 0$, and $0 \le \phi < 2\pi$. What is the smallest positive value of $\phi$ in radians? [b]p17.[/b] Find the sum of the distinct real roots of the equation $$\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.$$ [b]p18.[/b] If $a$ and $b$ satisfy the property that $a2^n + b$ is a square for all positive integers $n$, find all possible value(s) of $a$. [u]Round 7 [/u] [b]p19.[/b] Compute $(1 - \cot 19^o)(1 - \cot 26^o)$. [b]p20.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 5$, and $\angle ABC = 120^o$. Let point $E$ be any point inside $ABC$. The minimum of the sum of the squares of the distances from $E$ to the three sides of $ABC$ can be written in the form $a/b$ , where a and b are natural numbers such that the greatest common divisor of $a$ and $b$ is $1$. Find $a + b$. [b]p21.[/b] Let $m \ne 1$ be a square-free number (an integer – possibly negative – such that no square divides $m$). We denote $Q(\sqrt{m})$ to be the set of all $a + b\sqrt{m}$ where $a$ and $b$ are rational numbers. Now for a fixed $m$, let $S$ be the set of all numbers $x$ in $Q(\sqrt{m})$ such that x is a solution to a polynomial of the form: $x^n + a_1x^{n-1} + .... + a_n = 0$, where $a_0$, $...$, $a_n$ are integers. For many integers m, $S = Z[\frac{m}] = \{a + b\sqrt{m}\}$ where $a$ and $b$ are integers. Give a classification of the integers for which this is not true. (Hint: It is true for $ m = -1$ and $2$.) PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle intersects square $ABCD$ at points $A, E$, and $F$, where $E$ lies on $AB$ and $F$ lies on $AD$, such that $AE + AF = 2(BE + DF)$. If the square and the circle each have area $ 1$, determine the area of the union of the circle and square.

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $ 247,$ $ 275,$ and $ 756$ and end with the term $ 824.$ Let $ \mathcal{S}$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $ \mathcal{S}?$ $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$

$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE$, $CF$. Prove that the incenters of triangles $AGF$, $BHF$, $CHE$, $DGE$ lie on a circle. Proposed by Le Viet An (Vietnam)

Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.

Two circles \(\Gamma\) and \(\Sigma\) intersect at two distinct points \(A\) and \(B\). A line through \(B\) intersects \(\Gamma\) and \(\Sigma\) again at \(C\) and \(D\), respectively. Suppose that \(CA=CD\). Show that the centre of \(\Sigma\) lies on \(\Gamma\).